Question

How do you find the exact values of tan 7pi/8 using the half angle formula?

Answer 1

Find `tan ((7pi)/8)`

Ans: (1 - sqrt2)

Explanation:

Call `tan ((7pi)/8) = tan t`
`tan 2t = tan ((14pi)/8) = tan ((7pi)/4) = - tan pi/4 = - 1`
Apply the trig identity: `tan 2t = (2tan t)/(1 - tan^2 t)`
`tan 2t = - 1 = (2tan t)/(1 - tan^2 t)`
`tan^2 t - 2tan t - 1 = 0`
`D = d^2 = b^2 - 4ac = 4 + 4 = 8`--> `d = +- 2sqrt2.`
`tan t = 2/2 +- 2sqrt2/2 = 1 +- sqrt2`
Since the arc (7pi)/8 is located in Quadrant II, then the answer is:
`tan t = tan ((7pi)/8) = 1 - sqrt2`= - 0.414
Check by calculator.
`(7pi)/8 = 157.5` deg --> tan (157.5) = -0.414. OK